[Paper Review] Nonparametric Bayesian Deconvolution of a Symmetric Unimodal Density

Ya Su, Anirban Bhattacharya+3 more|arXiv (Cornell University)|Feb 17, 2020

Gene expression and cancer classification31 references4 citations

TL;DR

This paper proposes a nonparametric Bayesian deconvolution method for estimating a symmetric, unimodal density from heteroscedastic measurement error data, leveraging a Dirichlet process mixture of Gamma distributions to model the mixing density of symmetric uniform components. The method ensures shape constraints are preserved and demonstrates superior accuracy over standard kernel deconvolution in simulations and real GWAS/microarray data, with consistent estimation and scalable computation.

ABSTRACT

We consider nonparametric measurement error density deconvolution subject to heteroscedastic measurement errors as well as symmetry about zero and shape constraints, in particular unimodality. The problem is motivated by applications where the observed data are estimated effect sizes from regressions on multiple factors, where the target is the distribution of the true effect sizes. We exploit the fact that any symmetric and unimodal density can be expressed as a mixture of symmetric uniform densities, and model the mixing density in a new way using a Dirichlet process location-mixture of Gamma distributions. We do the computations within a Bayesian context, describe a simple scalable implementation that is linear in the sample size, and show that the estimate of the unknown target density is consistent. Within our application context of regression effect sizes, the target density is likely to have a large probability near zero (the near null effects) coupled with a heavy-tailed distribution (the actual effects). Simulations show that unlike standard deconvolution methods, our Constrained Bayesian Deconvolution method does a much better job of reconstruction of the target density. Applications to a genome-wise association study (GWAS) and microarray data reveal similar results.

Motivation & Objective

• To estimate the true distribution of effect sizes in genome-wide association studies (GWAS) and microarray data, where observed effect sizes are contaminated by measurement error.
• To enforce shape constraints—symmetry about zero and unimodality—on the deconvolved density, reflecting biological plausibility of effect size distributions.
• To handle heteroscedastic measurement errors, common in high-throughput biological data, where error variances vary across SNPs or genes.
• To develop a scalable Bayesian method that ensures consistent estimation of the target density under shape constraints.
• To outperform standard nonparametric deconvolution methods in reconstructing densities with a sharp peak near zero and heavy tails.

Proposed method

• Represents any symmetric, unimodal density as a mixture of symmetric uniform densities, exploiting a known representation theorem from Feller (1971).
• Models the mixing density using a Dirichlet process mixture of Gamma distributions to ensure smoothness and large support over the space of smooth densities.
• Implements a scalable Gibbs sampler for posterior computation, linear in sample size, enabling efficient inference on large datasets.
• Incorporates heteroscedastic measurement errors by allowing individual error variances σi² to vary across observations, as in GWAS with varying SNP imputation quality.
• Uses a Bayesian nonparametric framework to jointly estimate the mixing distribution and the target density, preserving shape constraints a priori.
• Employs a nonparametric prior on the mixing measure to allow flexibility while ensuring the resulting density is symmetric and unimodal.

Experimental results

Research questions

• RQ1 Can a Bayesian nonparametric deconvolution method effectively reconstruct a symmetric, unimodal density when measurement errors are heteroscedastic?
• RQ2 How does incorporating shape constraints—symmetry and unimodality—affect the accuracy of deconvolution compared to unconstrained methods?
• RQ3 Does the proposed method outperform standard kernel deconvolution in estimating effect size distributions with a sharp peak near zero and heavy tails?
• RQ4 Is the proposed method consistent under regularity conditions, even when the true density is not in the model's support?
• RQ5 Can the method be efficiently scaled to large-scale genomics data, such as GWAS with tens of thousands of SNPs?

Key findings

• In homoscedastic settings, the Constrained Bayesian method achieved a 46.8% lower integrated absolute error (IAE) than the kernel method at n=1000, with IAE of 0.730 vs. 1.069.
• At n=15000, the Constrained Bayes method reduced IAE to 0.474, while the kernel method remained near 1.000, indicating strong convergence and accuracy.
• In heteroscedastic settings, the Constrained Bayes method reduced IAE from 1.175 (kernel) to 0.532 (Constrained Bayes) at n=15000, showing robustness to varying error variances.
• The exceedance probability error was consistently lower for the Constrained Bayes method, with 0.159 at n=15000 in heteroscedastic settings, compared to 0.469 for the kernel method.
• The method successfully captured the sharp peak near zero and heavy-tailed nature of the true density, as seen in both simulated and real GWAS data.
• The proposed Gibbs sampler enabled linear-time computation, making the method practical for large-scale genomics applications.